where \(\Omega \)  is a bounded polyhedral domain in \(\mathbb {R}^d\)  (\(d=1,2,3\) ) with a piecewise smooth boundary \(\Gamma :=\partial \Omega \) ; \(u:\Omega \cup \Gamma \rightarrow \mathbb {R}\)  with \(u\in H_0^1(\Omega )\) ; \(q:\Omega \rightarrow \mathbb {R}\)  with \(q\in L^{\infty }(\Omega )\) ; and \(f:\Omega \rightarrow \mathbb {R}\)  with \(f\in H^{-1}(\Omega )\)  is given. The elliptic equation (REF ) describes various physical phenomena such as the flow of a fluid through some medium with the permeability \(q(x)\)  and the concentration \(u(x)\) , and the heat transfer in a material with the conductivity \(q(x)\)  and the temperature \(u(x)\) . For the diffusion coefficient \(q(x)\) , it is often impractical to measure it directly (e.g., when it is the conductivity of a medium), but it is easier to observe the solution \(u\)  of (REF ) or its gradient [1]}, [2]}. Hence, it is interesting to consider the inverse problem of identifying the diffusion coefficient \(q(x)\)  with observation data of the solution \(u\)  of the elliptic equation (REF ) or its gradient. This inverse problem finds applications in various industrial areas such as reservoir simulations, underground water investigations, geophysics and electrical impedance tomography. We refer to the monographs [3]}, [4]} for more introductions. Note that the elliptic equation (REF ) is linear if \(q(x)\)  is known, but the inverse problem of identifying \(q(x)\)  is nonlinear.
Also, as mentioned in [4]}, [6]}, \(q(x)\)  cannot be uniquely determined by \(u(x)\)  since \(q(x)\)  can be arbitrary when \(u\)  is constant on some open subset of \(\Omega \) . Thus, identifying the diffusion coefficient \(q(x)\)  of (REF ) is an ill-posed inverse problem.
